Transaction limits start when the fantasy season starts. So if you make moves in the offseason (i.e., before Week 1), they will not count. You can see the exact dates of your league's season by going to Team » Schedule.

The only roster moves that count against transaction limits are Adds and Claims. Anything else will not count. So if you cut a player, trade for a player, or if the commish imports a player, it will not count against your transaction limit.

We don't offer a mechanism currently to reset the transaction counter. As a workaround, you could offer owners bonus transactions by using League » Settings » Edit Rosters to perform their next couple of moves. For example, if they are owed two transactions, you could make their next two roster additions via Edit Rosters which won't cost them a transaction like it would if they did it themselves.